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The following theorem is known as Kugelsatz:

Let $X$ be an open set in $\mathbb[C}^n, \mathbb{C}^n, \quad n \geq 2$ and $K \subset X$ a compact subset such that $X\setminus K$ is connected. Then the restriction map $\rho: \mathcal{O}(X} mathcal{O}(X) \mapsto \mathcal{O}(X \setminus K)$ is an isomprphism of $\mathbb{C}$-algebras (this version after: Volker Scheideman, Introduction to Complex Analysis in Several Variables, Birkh\"{a}user 2005).

The first result of this kind is due to Hartogs, with $X$ and $K$ being concentric euclidean balls, hence the name (Kugel=ball). many texts in several complex variables have been written by German-speaking authors (Grauert+Fritzsche, Kaup brothers are other examples), so the German name stuck even in the English version. The theorem is also referred to as "tomato can principle".

Let $X$ be an open set in $\mathbb[C}^n, \quad n \geq 2$ and $K \subset X$ a compact subset such that $X\setminus K$ is connected. Then the restriction map $\rho: \mathcal{O}(X} \mapsto \mathcal{O}(X \setminus K)$ is an isomprphism of $\mathbb{C}$-algebras (this version after: Volker Scheideman, Introduction to Complex Analysis in Several Variables, Birkh\"{a}user 2005).
The first result of this kind is due to Hartogs, with $X$ and $K$ being concentric euclidean balls, hence the name (Kugel=ball). many texts in several complex variables have been written by German-speaking authors (Grauert+Fritzsche, Kaup brothers are other examples), so the German name stuck even in the English version. The theorem is also referred to as "tomato can principle".